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Stars: Their Properties. T. K. Prasad (Adapted from a lecture by Daniel Wang of UMass). Twinkle, twinkle, little star, How I wonder what you are. Up above the world so high, Like a diamond in the sky. Stars. Are Stars similar to our Sun?

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stars their properties

Stars: Their Properties

T. K. Prasad

(Adapted from a lecture by Daniel Wang of UMass)


Twinkle, twinkle, little star,How I wonder what you are.Up above the world so high,Like a diamond in the sky.


Are Stars similar to our Sun?

How far away are they?

Where did they come from?

What do they do?

Do they live forever?

the four basic parameters of a star
The Four Basic Parameters of a Star
  • Luminosity
  • Size
  • Mass
  • Surface Temperature

To infer these parameters, we need to know the distance!

luminosity and apparent brightness
Luminosity and Apparent Brightness
  • Luminosity is the measure of energy radiated by a star per second over all wavelengths. (Cf. Visual Luminosity)
    • Luminosity depends both on temperature and surface area. 
    • This cannot be determined by direct observation.
  • Apparent brightness is the amount of energy coming from the star per square meter per second, as measured on Earth. (cf. Flux)
      • This can be determined by direct observation.
cont d
  • Luminosity is an intrinsic property of a star, while apparent brightness depends on the distance to the observer.
  • Luminosity is how bright a star really is, while apparent brightness is how bright a star appears to an observer.
distances by triangulation
Distances by Triangulation

We can measure distances by comparing the position of objects observed from two ends of the “baseline” of a triangle.

  • Hold your thumb up, steadily in front of you.
  • Move your head from side to side and note the shift of your thumb with respect to background objects—this angular shift is called parallax.
  • Now look at your thumb while keeping your head steady but first closing one eye then the other.
  • Move your thumb closer to you—does it shift more or less with respect to the background?
  • You use parallax constantly to estimate distances.
  • Close your eyes. Have a neighbor dangle a pen in front of you, then open just one eye. Without moving your head, bring your hand in from the side and try to touch the pencil with just the tip of another pen.
  • Your brain processes the information from each eye and compares the angles to allow you to judge distances.
the geometry of parallax






The Geometry of Parallax


We use the Earth’s whole orbit as our baseline.

1 (AU)

D (in Parsecs) =

P (in arcseconds)

1 parsec (pc) = 3.26 ly. Other useful units: kpc and Mpc

parallax from a different planet
Parallax from a Different Planet

If we lived on Mars, orbiting 1.5 times farther away from the Sun, the parallax would be

  • the same as from Earth
  • 1.5 times smaller than from Earth
  • 1.5 times bigger than from Earth
digression proper motion of stars very slow

100,000 yrs ago

Digression:Proper Motion of Stars (Very Slow)


Surprising Fact:

It is easier to measure radial velocity using Doppler Effect than

transverse velocity!

100,000 yrs in future

Big Dipper

stellar parallax
Stellar Parallax
  • Since ancient Greek times, astronomers expected that if the Earth moved through space, we would see the stars shifting due to parallax.
    • If the Copernican model is correct, parallax of stars was a necessary consequence, but it was undetected until the 1830’s because of the huge distances of stars.
  • The nearest stars shift by only about 0.7 arcsec

1 / 0.7 = 1.4 parsec

This is about 4.3 light years

or about 27,000,000,000,000 miles !

survey question stellar parallax
Survey Question: Stellar Parallax

Suppose a star has a parallax of 0.01 arc seconds. How many parsecs away is it?

distance (in parsecs) = 1 / parallax (in arcsec)

Answer: 100

brightness distance and luminosity
Brightness, Distance, and Luminosity

L=4D2 B

apparent brightness



apparent brightness vs luminosity






Apparent Brightness vs Luminosity
  • Luminosity depends on Brightness & Distance

A appears brighter

b = L / 4πd2

A appears brighter

how to measure the surface temperature of a star
How to measure the surface temperature of a star?

Overall spectral shape (the peak of the blackbody continuous spectrum) is related to its temperature by

Wien’s Displacement Law:

T =2.9 × 106 K

λmax (nm)

More accurately, spectroscopically

Wein’s LawPeak frequency of radiation from a (star) blackbody is proportional to its (surface) temperature
spectral types of stars
Spectral Types of Stars
  • Spectral types are defined by the:
    • existence of absorption lines belonging to various elements, ions, & molecules in a star’s spectrum
    • the relative strengths of these lines
  • However, spectral type is not determined by a star’s composition.
    • all stars are made primarily of Hydrogen & Helium
reason for spectral types
Reason for Spectral Types
  • Spectral type is determined by a star’s surface temperature.
  • temperature dictates the energy states of electrons in atoms
  • temperature dictates the types of ions or molecules which exist
  • this, in turn, determines the number and relative strengths of absorption lines in the star’s spectrum
spectral type classification system
Spectral Type Classification System

(L T)


Oh Be A Fine Girl/Guy, Kiss Me!

50,000 K 3,000 K


Other Mnemonics: e.g., Officially, Bill always felt guilty kissing Monica Lewinsky tenderly

stellar size
Stellar Size
  • Stars are spherical so we characterize a star’s size by its radius.

Stellar Radii vary in sizefrom ~1500 RSun for a large Red Giant to 0.008 RSun for a WhiteDwarf.


How do we determine the radius of a star?

angular radius of star
Angular Radius of Star

The angular radius of the Sun is about 103 arc seconds. If another star like the Sun was 5 parsecs away (about 106 AU), what would its angular radius be?

  • 109 arc seconds
  • 100 arc seconds
  • 10-3 arc seconds
  • 10-9 arc seconds
temperature luminosity and size pulling them all together
Temperature, Luminosity, and Size – pulling them all together

A star’s luminosity, surface temperature, and size are all related by the Stefan-Boltzmann Law:

Stefan-Boltzmann Law

L=4πR2 σT4





L=4πR2 σT4

Two stars have the same surface temperature, butthe radius of one is 10 times the radius of the other.The larger star is

1) 10 times more luminous

2) 100 times more luminous

4) 1/10th as luminous

5) 1/100th as luminous


L=4πR2 σT4

L=4πD2 B

Suppose two stars are at equal distance and have the sameradius, but one has a temperature that is twice as great as theother. The apparent brightness of the hotter star is ____ as the other.

1) 1/2 as great

2) 1/4 as great

4) 4 times

5) 16 times as great

measurements of star properties
Measurements of Star Properties

Direct measurent


Distance + apparent brightness

( L=4D2 B)

Spectral type (or color)

Luminosity + temperature

(L=4R2 T4)

Apparent brightness (B)





Luminosity and temperature are the two independent intrinsic parameters of stars.

how do you weigh a star
How do you weigh a star?
  • Mass is the single most important property in how a star’s life and death will proceed.
  • The mass of a star can only be measured directly by observing the effect of its gravity on another object
  • This is most easily done for two stars which orbit one another --- a binary star!
newton s version of kepler s third law
Newton’s Version of Kepler’s Third Law

Star A

Newton was able to derive Kepler’s Third Law from his own Law of Gravity. In its most general form:


P2 (mA + mB)= a3

The orbital period of two objects (P) depends on the distance between them (a) and the sum of the masses of both objects (mA + mB).

So if P and a can be measured, mA + mB can be estimated.

Star B

Each star in a binary system moves in its own orbit around the system's center of mass.

orbits and masses of binaries
Orbits and Masses of Binaries

The primary importance of binaries is that they allow us to measure stellar parameters (especially mass).

We get the sum of the masses unless we see both stars moving.


Visual Binaries

But for most binaries, one cannot separate the stars even with most powerful telescopes. For them, we need to use the spectroscopic information.

visual binary star images

Sirius – the brightest

star in the sky.

Visual Binary Star Images

Mizar – in the handle of the

Big Dipper.

Albireo –

The “Cal” star


Spectroscopic Binaries





  • The total spread (size) of the Doppler shift gives velocities about center of mass  orbit sizes, a
  • The time to complete one repeating pattern  period, P


Recall: Doppler Shift tells only if it is moving toward or away


Eclipsing Binaries

  • Binary orbiting edge-on to our line of sight.
  • The stars alternately eclipse each other changing the apparent brightness.

From the eclipse duration, and orbital speed, we can also find the size of the star.

Thus one typically can tightly constrain the star masses in eclipsing binaries.

in review
In Review
  • There are four principal characteristics of a star:
    • Luminosity
    • Surface Temperature
    • Size
    • Mass

How may we classify stars?

We can take a census of stars and see what’s out there.

But first, let’s do some sociology in the classroom.

star a has a parallax that is twice that of star b what is the relationship between their distances
Star A has a parallax that is twice that of Star B. What is the relationship between their distances?
  • Star A is closer than Star B
  • Star B is closer than Star A
  • The stars are at the same distance
  • Not enough information is given
discussion question
Discussion Question

Make a plot that shows the generalrelationship between height and weight for humans.

- now add to your plot the population of basketball players who are very tall and very thin.

- now add the population of obese wrestlers

how can we classify stars
How can we classify stars
  • Collect information ona large sample of stars.
  • Measure their luminosities(need the distance!)
  • Measure their surface temperatures(need their spectra)
the hertzsprung russell diagram
The Hertzsprung-Russell Diagram

Around 1910, Ejnar Hertzsprung (Dane) and Henry Norris Russell (American) had the idea of plotting the luminosity of a star against its spectral type. For a star cluster, all the stars are at the same distance. So, apparent brightness vs spectral type is basically the same as luminosity vs temperature. They found that stars appeared only in certain parts of the diagram.

the hertzsprung russell diagram52
The Hertzsprung-Russell Diagram

The Main Sequence

  • ~90% of all stars are in the main sequence (MS)
  • ~90% of all MS stars are cooler spectral types than the Sun (i.e., at the lower MS)
  • All MS stars fuse H into He in their cores.
the hertzsprung russell diagram53

Mass of MS Star

L  M3.5

The Hertzsprung-Russell Diagram

Mass-Luminosity Relation:

  • L  M3.5
  • For example, if the mass of a star is doubled, its luminosity increases by a factor 23.5 ~ 11.

The relation is for main sequence stars only!

the hertzsprung russell diagram54
The Hertzsprung-Russell Diagram

Red Giants

- Red Giant starsare very large, cooland quite bright.

e.g., Betelgeuse is150,000 times moreluminous than the Sunbut is only 3,500K onthe surface. It’s radiusis 1,000 times that of the Sun.

the hertzsprung russell diagram56
The Hertzsprung-Russell Diagram

White Dwarfs

- White Dwarfsare hot, but sincethey are so small,they are not veryluminous.

main sequence lifetime
Main Sequence Lifetime
  • All M-S stars have temperatures sufficient to fuse H into He in their cores.
  • Luminosity depends directly on mass:
    • more mass = more pressure from upper layers
    • fusion rates must be high to maintain equilibrium
  • Lifetime  (Amt. of Fuel)/(Rate of Burning)  M / L  M / M3.5 1 / M2.5
  • Higher mass stars have shorter lives!
the hertzsprung russell diagram58

More mass,more fuel,very fast burning.

Less mass,less fuel,slow, steady burning.

The Hertzsprung-Russell Diagram


Lifetimeof Star



SUV vs Honda Civic

review questions the h r diagram
Review Questions: The H-R Diagram
  • Where are most stars?
  • What is the common characteristics of MS stars?
  • What determines the location of a star in the MS?
  • Where do you find the largest stars?
  • The smallest?
  • The most massive one?
  • The coolest stars?
  • How do we know the age of a star?

1. MS, 2. H He, 3. M, 4. upperright, 5. lowerleft, 6. upperleft, 7. lowerright, 8. normally we don’t

mass lifetime relation ms

fuel mass

  • Lifetime =

rate of consumption





~ M–2.5



Mass-Lifetime Relation (MS)
  • High mass stars have more fuel
  • but they burn it much faster

1 Myr

30 Myr

300 Myr



1 Gyr


Short lifetime



10 Gyr

O5 V (40Ms) : 1 Myr

G2 V (Sun) : 10 Gyr

M5 V (0.2 Ms) : 500 Gyr





60 Gyr



  • No star with M < 0.9Ms has yet died
  • (tUniverse = 13.7 Gyr)

Long Lifetime

e.g. for a 4 Msun star (e.g. Vega)

L = 43.5 = 128 Lsun

tlife = 4–2.5 = 0.03tsun = 300 Myr

aging of a cluster of stars
Aging of a cluster of stars
  • MS lasts until H is exhausted in the core.
  • Clues to the next stage are visible in older star clusters.
  • The brightest stars are gone, replaced by red giants.
why are clusters useful to astronomers
Why are clusters useful to astronomers?
  • All stars in a cluster are at about same distance from Earth.
  • All stars in a cluster are of about the same age.
  • Clusters therefore are natural laboratory in which mass, rather than age, of stars is only significant variable.
the hertzsprung russell diagram67

All these stars in the cluster have burned themselves out!

The Hertzsprung-Russell Diagram

We can date a cluster by observing itspopulation ofstars.

The oldest clustersknown have beenmeasured to be ~13 billion years old.



Anatomy of a Main Sequence Star

Hydrogenburning core



up the red giant branch
Up the red giant branch

As hydrogen in the core is being used up, it starts to contract, raising temperature in the surrounding. Eventually, hydrogen will burn only in a shell. There is less gravity from above to balance this pressure. The Sun will then swell to enormous size and luminosity, and its surface temperature will drop,  a red giant.

Sun in ~5 Gyr

Sun today

helium fusion at the center of a giant
Helium fusion at the center of a giant
  • While the exterior layers expand, the helium core continues to contract, while growing in mass, and eventually becomes hot enough (100 million Kelvin) for helium to begin to fuse into carbon
  • Carbon ash is deposited in core and eventually a helium-burning shell develops. This shell is itself surrounded by a shell of hydrogen undergoing nuclear fusion.
  • For a star with M< 1 Msun, the carbon core never gets hot enough to ignite nuclear fusion.
  • In very massive stars, elements can be fused into Fe.

The Sun will expand and cool again, becoming a red giant. Earth will be engulfed and vaporized within the Sun. The Sun’s core will consist mostly of carbon.

  • Red Giants create most of the Carbon in the universe (from which organic molecules—and life—are made)
how can two stars have the same temperature but vastly different luminosities
How can two stars have the same temperature, but vastly different luminosities?
  • The luminosity of a star depends on 2 things:
    • surface temperature
    • surface area (radius)

L =  T4 4  R2

  • The stars have different sizes!!
  • The largest stars are in the upper right corner of the H-R Diagram.

L =  T4 4  R2. If Star A is twice as hot and one fourth the radius of Star B, then it should be…

(1) 1/4 as luminous as Star B

(2) just as luminous as Star B

(3) 16 times as luminous as Star B

(4) 64 times as luminous as Star B

review the hertzprung russell h r diagram
Review: The Hertzprung-Russell (H-R) Diagram
  • One of the most important diagrams in astronomy for star classification
  • Spectral Types of Stars
    • OBAFGKM: O=hottest, M=coolest. 
    • spectral type carries almost same info as color or temperature
h r diagram main sequence
H-R Diagram: main sequence
  • Most (about 90%) stars -- including the Sun -- appear to lie on the main sequence (MS).
  • Mass determines location of star on MS:
  • L  M3.5 -- Luminosity depends very strongly on mass.
  • The defining characteristic of a MS star is that it is fusing H to He.
other stars
Other stars
  • Red Giants are bright because they're big, even though cool.
    • Appear near the upper right section of the HR Diagram.
    • They have a bigger radius than the stars of the same temperature which gives them a higher luminosity.
  • White Dwarf's are faint because they're tiny, even though hot.
    • Appear in the lower left section of the HR Diagram.
    • They are extremely hot, yet appear very dim due to their extremely small size.
why are there no main sequence m class stars visible to the naked eye
Why are there no Main Sequence M-class stars visible to the naked eye?

(1) they are very rare and all very far away.

(2) they are so cool that they only emit in the infrared.

(3) they are too dim to be seen even if they are only a few light years away.